A method discovered by
Fermat for proving the
non-existence of numbers bearing certain properties which are bounded by a closed system (this is most often the
natural numbers).
The quick and dirty of it is, you suppose the existence of a set of numbers which fit an equation, and go about proving that their existence implies the existence of another set of numbers which meet the same parameters, and all of which are smaller than their counterparts in the first set. And it follows that since this new set meets the same requirements as the first set, its existence implies another smaller set. And another and another and so on. The trick is, since the solutions are restricted to a closed set, eventually you will imply the existence of solutions outside that set... and wind up with something like a square number less than zero, which is a contradiction. Because that contradiction stems solely from the assumption that there is a solution to a certain equation, you have proven that there is no solution, no numbers to meet those parameters.
We can use Infinite Descent to prove such useful things as that the square root of a prime number is irrational. Fermat used it to prove his famous Last Theorem for the specific case of n = 4. Note that you can use Infinite Ascent if you're dealing with solutions that would be limited by an upper bound.